Erdős Problems

Tags Prizes

0 solved out of 3 shown

OPEN

Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$.

In [Er81d] Erdős proved that $\gg n$ many circles is possible, and that there cannot be more than $n(n-1)$ many circles. Elekes [El84] has a simple construction of a set with $\gg n^{3/2}$ such circles. This may be the correct order of magnitude.

In [Er75h] Erdős also asks how many such unit circles there must be if the points are in general position.

See also [506] and [831].

OPEN

Let $n_k$ be minimal such that if $n_k$ points in $\mathbb{R}^2$ are in general position then there exists a subset of $k$ points such that all $\binom{k}{3}$ triples determine circles of different radii.

Determine $n_k$.

#827: [Er75h][Er78c]

geometry

In [Er75h] Erdős asks whether $n_k$ exists. In [Er78c] he gives a simple argument which proves that it does, and in fact \[n_k \leq k+2\binom{k-1}{2}\binom{k-1}{3}.\]

OPEN

Let $h(n)$ be maximal such that in any $n$ points in $\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many circles of different radii passing through three points. Estimate $h(n)$.

#831: [Er75h]

geometry